Transient buoyant convection in a square enclosure of a non-Newtonian fluid is studied. A simple power-law fluid model, with the power-law index n and the consistency coefficient K, is adopted. Flow is initiated from the motionless isothermal initial state by abruptly raising the temperature at one vertical sidewall and lowering the temperature at the opposite vertical sidewall. The appropriately defined overall Rayleigh number Ra is large to render a boundary layertype flow. A scale analysis is performed to gain a rudimentary understanding of the evolution process. Principal force balances are considered in each of the transient stages. Comprehensive numerical solutions are acquired to the governing equations. The transient flow and thermal characteristics, both in the boundary layers and in the interior, are portrayed. The effects of Ra and of n on the behavior of the Nusselt number are delineated. The system-wide heat transport at the steady state is calculated. Based on the numerical results, the Nusselt number correlations are proposed. The numerical solutions are in broad qualitative agreement with the descriptions obtainable from the scale analysis.